Subgroups — C₃×D₄×Dic₁₅ #13088028

Order 1440.

430 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₁₅	C₂	1	2	yes	maximal, complemented
3	C₃×#6719	C₂	1	2	yes	maximal, complemented
4	C₃×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×Dic₁₅	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃×D₄×Dic₅	—	3	1	no	maximal
8	C₂³×C₃²×C₅	C₂²	1	2	yes	—
9	C₃²×C₅×D₄	C₄	1	4	yes	complemented
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×Dic₁₅	—	2	4	no	—
12	C₂×C₃×Dic₁₅	C₂²	1	2	yes	—
13	C₂×C₃×Dic₁₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×Dic₁₅	C₂²	1	1	yes	characteristic
15	C₃×D₄×Dic₃	—	5	1	no	maximal, Hall
16	C₂×C₃×C₅×D₄	—	2	1	no	—
17	C₂×C₃×C₅×D₄	D₃	1	1	yes	characteristic
18	C₂×C₃×C₅×D₄	C₂×C₃	1	1	yes	characteristic
19	C₂²×Dic₁₅	C₂×C₃	1	2	yes	complemented
20	C₂²×C₃×Dic₅	—	3	2	no	—
21	#6719	C₂×C₃	1	2	yes	complemented
22	C₄⋊Dic₁₅	C₂×C₃	1	1	yes	characteristic, complemented
23	C₄×Dic₁₅	C₂×C₃	1	1	yes	characteristic, complemented
24	C₃×C₂²⋊Dic₅	—	3	2	no	—
25	C₃×C₄⋊Dic₅	—	3	1	no	—
26	C₃×C₄×Dic₅	—	3	1	no	—
27	C₂²×C₃²×C₅	—	2	2	no	—
28	C₂²×C₃²×C₅	—	2	2	no	—
29	C₂²×C₃²×C₅	C₂×C₄	1	4	yes	complemented
30	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
31	C₃²×C₄×C₅	C₂×C₄	1	2	yes	complemented
32	C₃×Dic₁₅	—	2	2	no	—
33	C₃×Dic₁₅	—	2	1	no	—
34	C₃×Dic₁₅	D₄	1	2	yes	direct product factor
35	D₄×Dic₅	—	3	1	no	Hall
36	C₂×C₃²×D₄	D₅	1	1	yes	characteristic
37	C₂²×C₃×Dic₃	—	5	2	no	—
38	C₃×#217	—	5	2	no	—
39	C₃×C₄⋊Dic₃	—	5	1	no	—
40	C₃×C₄×Dic₃	—	5	1	no	—
41	C₂³×C₃×C₅	—	2	2	no	—
42	C₂³×C₃×C₅	C₂×D₃	1	2	yes	—
43	C₂³×C₃×C₅	C₂²×C₃	1	2	yes	—
44	C₃×C₅×D₄	—	2	4	no	—
45	C₃×C₅×D₄	Dic₃	1	4	yes	complemented
46	C₃×C₅×D₄	C₃×C₄	1	4	yes	complemented
47	C₂×C₃×C₄×C₅	—	2	1	no	—
48	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
50	C₂×Dic₁₅	—	2	4	no	—
51	C₂×Dic₁₅	C₂²×C₃	1	1	yes	characteristic
52	C₂×Dic₁₅	C₂²×C₃	1	2	yes	—
53	C₂×Dic₁₅	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₃×Dic₅	—	3	1	no	—
55	C₂×C₃×Dic₅	—	3	1	no	—
56	C₂×C₃×Dic₅	—	3	2	no	—
57	C₂×C₃×Dic₅	—	6	4	no	—
58	C₃×C₄×D₄	—	15	1	no	—
59	D₄×Dic₃	—	5	1	no	—
60	C₂×C₃²×C₅	—	2	4	no	—
61	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃²×C₅	#41	1	1	yes	characteristic
64	C₂×C₅×D₄	C₃×D₃	1	1	yes	characteristic
65	C₂²×Dic₅	—	3	2	no	—
66	C₂²⋊Dic₅	—	3	2	no	—
67	C₄⋊Dic₅	—	3	1	no	—
68	C₄×Dic₅	—	3	1	no	—
69	C₂³×C₃²	C₂×D₅	1	2	yes	—
70	C₃²×D₄	Dic₅	1	4	yes	complemented
71	C₂×C₃²×C₄	C₂×D₅	1	1	yes	characteristic
72	C₂×C₃×Dic₃	—	5	1	no	—
73	C₂×C₃×Dic₃	—	5	1	no	—
74	C₂×C₃×Dic₃	—	5	2	no	—
75	C₂×C₃×Dic₃	—	10	4	no	—
76	C₂²×C₃×C₅	—	2	2	no	—
77	C₂²×C₃×C₅	—	2	2	no	—
78	C₂²×C₃×C₅	—	2	2	no	—
79	C₂²×C₃×C₅	—	2	2	no	—
80	C₂²×C₃×C₅	—	2	4	no	—
81	C₂²×C₃×C₅	—	2	1	no	—
82	C₂²×C₃×C₅	—	4	2	no	—
83	C₂²×C₃×C₅	—	4	2	no	—
84	C₂²×C₃×C₅	C₂×Dic₃	1	4	yes	complemented
85	C₂²×C₃×C₅	C₂×C₃×C₄	1	4	yes	complemented
86	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₃×C₄×C₅	—	2	2	no	—
89	C₃×C₄×C₅	C₂×Dic₃	1	2	yes	complemented
90	C₃×C₄×C₅	C₂×C₃×C₄	1	2	yes	complemented
91	Dic₁₅	—	2	2	no	—
92	Dic₁₅	—	2	1	no	—
93	Dic₁₅	C₃×D₄	1	2	yes	direct product factor
94	C₃×Dic₅	—	3	2	no	—
95	C₃×Dic₅	—	6	2	no	—
96	C₃×Dic₅	—	6	1	no	—
97	C₂×C₃×D₄	—	2	1	no	—
98	C₂×C₃×D₄	C₃×D₅	1	1	yes	characteristic
99	C₂×C₃×D₄	D₁₅	1	1	yes	characteristic
100	C₂²×C₃×C₄	—	15	2	no	—
101	C₂²×Dic₃	—	5	2	no	—
102	C₃×C₄⋊C₄	—	15	1	no	—
103	C₃×C₂²⋊C₄	—	15	2	no	—
104	C₃×C₄²	—	15	1	no	—
105	#217	—	5	2	no	—
106	C₄⋊Dic₃	—	5	1	no	—
107	C₄×Dic₃	—	5	1	no	—
108	C₃²×C₅	C₄×D₄	1	1	yes	characteristic, complemented, Hall
109	C₂³×C₅	C₂×C₃×D₃	1	2	yes	—
110	C₅×D₄	C₃×Dic₃	1	4	yes	complemented
111	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
112	C₂×Dic₅	—	3	1	no	—
113	C₂×Dic₅	—	3	1	no	—
114	C₂×Dic₅	—	3	2	no	—
115	C₂×Dic₅	—	6	4	no	—
116	C₂²×C₃²	—	2	2	no	—
117	C₂²×C₃²	—	2	2	no	—
118	C₂²×C₃²	C₂×Dic₅	1	4	yes	complemented
119	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
120	C₃²×C₄	C₂×Dic₅	1	2	yes	complemented
121	C₃×Dic₃	—	5	2	no	—
122	C₃×Dic₃	—	10	2	no	—
123	C₃×Dic₃	—	10	1	no	—
124	C₄×D₄	—	15	1	no	Sylow
125	C₂×C₃×C₅	—	2	4	no	—
126	C₂×C₃×C₅	—	2	4	no	—
127	C₂×C₃×C₅	—	2	1	no	—
128	C₂×C₃×C₅	—	2	1	no	—
129	C₂×C₃×C₅	—	2	1	no	—
130	C₂×C₃×C₅	—	4	4	no	—
131	C₂×C₃×C₅	D₃×D₄	1	1	yes	characteristic
132	C₂×C₃×C₅	#237	1	1	yes	characteristic
133	C₂×C₃×C₅	C₂²×Dic₃	1	1	yes	characteristic
134	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
137	C₂³×C₃	—	2	2	no	—
138	C₂³×C₃	C₂×C₃×D₅	1	2	yes	—
139	C₂³×C₃	C₂×D₁₅	1	2	yes	—
140	C₃×D₄	—	2	4	no	—
141	C₃×D₄	C₃×Dic₅	1	4	yes	complemented
142	C₃×D₄	Dic₁₅	1	4	yes	direct product factor
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	15	1	no	—
145	C₂×C₃×C₄	—	15	1	no	—
146	C₂×C₃×C₄	—	15	2	no	—
147	C₂×C₃×C₄	—	30	4	no	—
148	C₂×C₃×C₄	C₂×C₃×D₅	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×D₁₅	1	1	yes	characteristic
150	C₂×Dic₃	—	5	1	no	—
151	C₂×Dic₃	—	5	1	no	—
152	C₂×Dic₃	—	5	2	no	—
153	C₂×Dic₃	—	10	4	no	—
154	C₂²×C₅	—	2	2	no	—
155	C₂²×C₅	—	2	2	no	—
156	C₂²×C₅	C₂×C₃×Dic₃	1	4	yes	complemented
157	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
158	C₄×C₅	C₂×C₃×Dic₃	1	2	yes	complemented
159	Dic₅	—	3	2	no	—
160	Dic₅	—	6	2	no	—
161	Dic₅	—	6	1	no	—
162	C₂×C₃²	—	2	4	no	—
163	C₂×C₃²	D₄×D₅	1	1	yes	characteristic
164	C₂×C₃²	#711	1	1	yes	characteristic
165	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
166	C₂×D₄	C₃×D₁₅	1	1	yes	characteristic
167	C₂²×C₄	—	15	2	no	—
168	C₄⋊C₄	—	15	1	no	—
169	C₂²⋊C₄	—	15	2	no	—
170	C₄²	—	15	1	no	—
171	C₃×C₅	—	2	1	no	—
172	C₃×C₅	D₄×Dic₃	1	1	yes	characteristic, complemented
173	C₃×C₅	C₃×C₄×D₄	1	1	yes	characteristic, complemented
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	4	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	4	2	no	—
181	C₂²×C₃	—	4	2	no	—
182	C₂²×C₃	C₂×C₃×Dic₅	1	4	yes	complemented
183	C₂²×C₃	C₂×Dic₁₅	1	4	yes	complemented
184	C₂²×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
185	C₂²×C₃	C₂²×D₁₅	1	1	yes	characteristic
186	C₃×C₄	—	2	2	no	—
187	C₃×C₄	—	15	2	no	—
188	C₃×C₄	—	30	2	no	—
189	C₃×C₄	—	30	1	no	—
190	C₃×C₄	C₂×C₃×Dic₅	1	2	yes	complemented
191	C₃×C₄	C₂×Dic₁₅	1	2	yes	complemented
192	Dic₃	—	5	2	no	—
193	Dic₃	—	10	2	no	—
194	Dic₃	—	10	1	no	—
195	C₂×C₅	—	2	4	no	—
196	C₂×C₅	C₃×D₃×D₄	1	1	yes	characteristic
197	C₂×C₅	C₃×#237	1	1	yes	characteristic
198	C₂×C₅	C₂²×C₃×Dic₃	1	1	yes	characteristic
199	C₃²	D₄×Dic₅	1	1	yes	characteristic, complemented, Sylow
200	C₂³	C₂×C₃×D₁₅	1	2	yes	—
201	D₄	C₃×Dic₁₅	1	4	yes	direct product factor
202	C₂×C₄	—	15	1	no	—
203	C₂×C₄	—	15	1	no	—
204	C₂×C₄	—	15	2	no	—
205	C₂×C₄	—	30	4	no	—
206	C₂×C₄	C₂×C₃×D₁₅	1	1	yes	characteristic
207	C₂×C₃	—	2	4	no	—
208	C₂×C₃	—	2	4	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	—	2	1	no	—
212	C₂×C₃	—	4	4	no	—
213	C₂×C₃	C₃×D₄×D₅	1	1	yes	characteristic
214	C₂×C₃	C₃×#711	1	1	yes	characteristic
215	C₂×C₃	C₂²×C₃×Dic₅	1	1	yes	characteristic
216	C₂×C₃	D₄×D₁₅	1	1	yes	characteristic
217	C₂×C₃	#6819	1	1	yes	characteristic
218	C₂×C₃	C₂²×Dic₁₅	1	1	yes	characteristic
219	C₅	C₃×D₄×Dic₃	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	C₂×C₃×Dic₁₅	1	4	yes	complemented
223	C₂²	C₂²×C₃×D₁₅	1	1	yes	characteristic
224	C₄	—	15	2	no	—
225	C₄	—	30	2	no	—
226	C₄	—	30	1	no	—
227	C₄	C₂×C₃×Dic₁₅	1	2	yes	complemented
228	C₃	—	2	1	no	—
229	C₃	C₃×D₄×Dic₅	1	1	yes	characteristic, complemented
230	C₃	D₄×Dic₁₅	1	1	yes	characteristic, direct product factor
231	C₂	—	2	4	no	—
232	C₂	C₃×D₄×D₁₅	1	1	yes	characteristic
233	C₂	C₃×#6819	1	1	yes	characteristic
234	C₂	C₂²×C₃×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×D₁₅
Fitting, prime radical	C₂×C₃²×C₅×D₄	C₂
socle	C₂²×C₃²×C₅	C₂³
cosocle	C₂²×C₃×C₅	C₂³×C₃
hypercenter	C₂×C₃×D₄	D₁₅
hypocenter	C₃×C₅	C₃×C₄×D₄
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×D₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×C₃×Dic₁₅	C₂	semidirect
3. C₃×#6719	C₂	semidirect
4. C₃×C₄⋊Dic₁₅	C₂	semidirect
5. C₃×C₄×Dic₁₅	C₂	semidirect
6. D₄×Dic₁₅	C₃	direct (N × H)
9. C₃²×C₅×D₄	C₄	semidirect
19. C₂²×Dic₁₅	C₂×C₃	semidirect
21. #6719	C₂×C₃	semidirect
22. C₄⋊Dic₁₅	C₂×C₃	semidirect
23. C₄×Dic₁₅	C₂×C₃	semidirect
29. C₂²×C₃²×C₅	C₂×C₄	semidirect
31. C₃²×C₄×C₅	C₂×C₄	semidirect
34. C₃×Dic₁₅	D₄	direct (N × H)
45. C₃×C₅×D₄	Dic₃	semidirect
46. C₃×C₅×D₄	C₃×C₄	semidirect
70. C₃²×D₄	Dic₅	semidirect
84. C₂²×C₃×C₅	C₂×Dic₃	semidirect
85. C₂²×C₃×C₅	C₂×C₃×C₄	semidirect
89. C₃×C₄×C₅	C₂×Dic₃	semidirect
90. C₃×C₄×C₅	C₂×C₃×C₄	semidirect
93. Dic₁₅	C₃×D₄	direct (N × H)
108. C₃²×C₅	C₄×D₄	semidirect
110. C₅×D₄	C₃×Dic₃	semidirect
118. C₂²×C₃²	C₂×Dic₅	semidirect
120. C₃²×C₄	C₂×Dic₅	semidirect
141. C₃×D₄	C₃×Dic₅	semidirect
142. C₃×D₄	Dic₁₅	direct (N × H)
156. C₂²×C₅	C₂×C₃×Dic₃	semidirect
158. C₄×C₅	C₂×C₃×Dic₃	semidirect
172. C₃×C₅	D₄×Dic₃	semidirect
173. C₃×C₅	C₃×C₄×D₄	semidirect
182. C₂²×C₃	C₂×C₃×Dic₅	semidirect
183. C₂²×C₃	C₂×Dic₁₅	semidirect
190. C₃×C₄	C₂×C₃×Dic₅	semidirect
191. C₃×C₄	C₂×Dic₁₅	semidirect
199. C₃²	D₄×Dic₅	semidirect
201. D₄	C₃×Dic₁₅	direct (N × H)
219. C₅	C₃×D₄×Dic₃	semidirect
222. C₂²	C₂×C₃×Dic₁₅	semidirect
227. C₄	C₂×C₃×Dic₁₅	semidirect
229. C₃	C₃×D₄×Dic₅	semidirect
230. C₃	D₄×Dic₁₅	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×C₅×D₄	2
C₂²×C₃×Dic₁₅	2
C₃×#6719	2
C₃×C₄⋊Dic₁₅	2
C₃×C₄×Dic₁₅	2
D₄×Dic₁₅	3
C₃×D₄×Dic₅	3
C₃×D₄×Dic₃	5

Subgroups — C3×D4×Dic15 #13088028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×Dic₁₅ #13088028